The Coxeter transformation on Cominuscule Posets

Let $\mathsf{J(C)}$ be the poset of order ideals of a cominuscule poset $\mathsf{C}$ where $\mathsf{C}$ comes from two of the three infinite families of cominuscule posets or the exceptional cases. We show that the Auslander-Reiten translation $τ$ on the Grothendieck group of the bounded derived category for the incidence algebra of the poset $\mathsf{J(C)}$, which is called the \emph{Coxeter transformation} in this context, has finite order. Specifically, we show that $τ^{h+1}=\pm id$ where $h$ is the Coxeter number for the relevant root system.

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